# MAT 421 Complex Variables

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### Revised Course Description

This course covers the algebra and geometry of the complex numbers; point sets, sequences and mappings; analytic functions; elementary functions; differentiation; integration; power series; the calculus of residues; and applications.

MAT 421 meets for three hours of lecture per week.

### Prerequisites

Students must have earned at least a "C" in courses equivalent to MAT 213-Vector Calculus and MAT 271-Foundations of Higher Mathematics before enrolling in MAT 421.

### Objectives

After completing MAT 421 the student should

• understand complex numbers, the algebra and geometry of complex numbers and the complex plane
• understand and work with complex vectors, polar forms, powers, and roots
• understand limits and continuity, analyticity, and the Cauchy-Riemann equation
• understand complex exponential, trigonometric, hyperbolic, logarithmic and power functions
• understand complex integration, contour integrals, the Cauchy Integral Theorem and formula, and bounds for analytic functions
• understand sequences and series, including Taylor series, power series, and Laurent series and their use in representing analytic functions
• understand residue theory
• be able to prove basic theorems related to the above concepts
• apply mathematical reasoning and the theory of complex variables to solve theoretical and applied problems.

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 421.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of topics that are covered in MAT 421.

### Text

Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, by Saff and Snider.

• Chapter 1-Complex Numbers
• The Algebra of Complex Numbers
• Point Representation, Absolute Value, and Complex Conjugates
• Vectors, Powers, Roots, and Polar Forms
• Chapter 2-Analytic Functions
• Functions of a Complex Variable
• Limits, Continuity, and Analyticity
• The Cauchy-Reimann Equation and Harmonic Functions
• Chapter 3-Elementary Functions
• Exponential, Trigonometric, Hyperbolic, and Logarithmic Functions
• Complex Powers and Inverse Trigonometric Functions
• Chapter 4-Complex Integration
• Contours and Contour Integrals
• Independence of Path
• Cauchy's Integral Theorem including the Deformation of Contours Approach and the Vector Analysis Approach
• Cauchy's Integral Formula and Bounds for Analytic Functions
• Chapter 5-Series Representations for Analytic Functions
• Sequences and Series, including Taylor, Power, and Laurent Series
• Zeros, Singularities, and the Point at Infinity
• Chapter 6-Residue Theory
• The Residue Theorem
• Trigonometric Integrals, Improper Integrals, and Integrals involving Multiple-Valued Functions

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 421. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.